Singular elliptic problems with unbalanced growth and critical exponent

Abstract

In this article, we study the existence and multiplicity of solutions of the following (p, q)-Laplace equation with singular nonlinearity:where Ω is a bounded domain in R n with smooth boundary, 1 < q < p < r p * , where p * = np n−p , 0 < δ < 1, n > p and λ, β > 0 are parameters. We prove existence, multiplicity and regularity of weak solutions of (P λ ) for suitable range of λ. We also prove the global existence result for problem (P λ ).

Existence of multiple positive solutions for singular p‐q‐Laplacian problems with critical nonlinearities

“…And the existence results in all of for quasilinear problems with critical term can be seen in Li and Yang 20 . On the basis of the p‐q ‐Laplacian equation with singular term, the critical term was added to above equation in Kumar et al 21 That is to consider the equation where Ω is a bounded domain in with smooth boundary. 1 < q < p < r ≤ p ∗ , where and λ , β > 0 are parameters.…”

On double phase Kirchhoff problems with singular nonlinearity

“…It turns out that the global minimizers of J λ restricted to two of them are the solutions we seek and the third one is the empty set for small values of the parameter λ > 0. This method has become a very powerful tool and has been used, for example, in the works of Chen-Kuo-Wu [9] (for degenerate Kirchhoff Laplacian problems with sign-changing weight), Fiscella-Mishra [22] (for fractional Kirchhoff problems), Kumar-Rȃdulescu-Sreenadh [32] (for critical (p, q)-equations), Liao-Zhang-Liu-Tang [33] (for Kirchhoff Laplacian problems), Mukherjee-Sreenadh [39] (for fractional problems), Papageorgiou-Repovš-Vetro [41] (for (p, q)-equations), Papageorgiou-Winkert [42] (for p-Laplacian problems), Tang-Chen [46] (for ground state solutions of Schrödinger type), see also the references therein.…”

Nodal solutions for double phase Kirchhoff problems with vanishing potentials

“…Usually, the reaction term c(x, u) is a polynomial of u with variable coefficient (see [27]). This kind of problem has been widely investigated by many authors, see for instance [27,31,32,39,[42][43][44]46] and the references therein. In particular, in [17], using a minimization argument and a quantitative deformation lemma, the authors proved the existence of nodal solutions for the following class of (p, q) problems − div(a(|∇u| p )|∇u| p−2 ∇u) + V (x)b(|u| p )|u| p−2 u = K(x)f (u) in R N , where N ≥ 3, 2 ≤ p < N , a, b, f ∈ C 1 (R), and V, K are continuous and positive functions (see also [16]).…”